On Toughness and Hamiltonicity of 2K₂‐Free Graphs

Abstract

The toughness of a (noncomplete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields |A|/t components. Determining toughness is an NP‐hard problem for general input graphs. The toughness conjecture of Chvátal, which states that there exists a constant t such that every graph on at least three vertices with toughness at least t is hamiltonian, is still open for general graphs. We extend some known toughness results for split graphs to the more general class of 2K2‐free graphs, that is, graphs that do not contain two vertex‐disjoint edges as an induced subgraph. We prove that the problem of determining toughness is polynomially solvable and that Chvátal's toughness conjecture is true for 2K2‐free graphs.

Nonempty Intersection of Longest Paths in $2K_2$-Free Graphs

“…By lemma 2.2 (3,4), S is an independent set. Let T = V (P ) − S. By Lemma 2.2 (1,2), V (P ) contains at least 2k + 1 vertices.…”

Edge-dominating cycles, k-walks and Hamilton prisms in 2K2-free graphs

“…Considerable work went into investigating this computational problem for various classes of graphs. In particular, recently, Broersma, Patel and Pyatkin proved [6] that toughness of a 2K 2 -free graph, i.e. a graph that does not contain an induced copy of the disjoint union of two edges, can be found in polynomial time.…”

“…figure on the right, where we have a 2K 2 -free graph with a 2-walk, but without Hamiltonian prism. It is worth mentioning that the toughness constant in Theorem 2 is much better than 3 2 , the lower bound on toughness of a 2K 2 -free graph needed for its Hamiltonicity, see [6,Sect. 4].…”

“…Suppose G is a triangle-free 2K 2free graph. By[6, Theorem 4], if |V (G)| ≥ 3 then G is Hamiltonian,3 and so3 We only need to rely on[6, Theorem 4] for the case of G not having a dominating cycle C of even length, for otherwise we have case 1 of Lemma 2 at our disposal; |C| can be odd only in the case of G not bipartite; such a G is, by [6, Lemma 2], of C * 5 -type, i.e. it has a rather special structure.…”

Degree, toughness and subgraph conditions for hamiltonian properties of graphs

“…They are hamiltonian (even if the sets I v have different cardinalities, as long as the graphs are 1-tough. We refer to [16], where these graphs are called C * 5 -type graphs and treated as special cases of triangle-free 2K 2free graphs). Using that C 5 is K 1 ∪ P 4 -free, it is easy to check that all these graphs are K 1 ∪ P 4 -free.…”